A More General Optimization Problem for Uniquely Decodable Codes

• Ahmed A. Belal
Conference paper
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)

Abstract

The problem of finding the integers r (i), i = 1,2,.....,N that minimize the cost function $${\text{J}}\;{\text{ = }}\;\sum\limits_{{\text{i = 1}}}^{\text{N}} {\text{A}} \;\left( {\text{i}} \right)\;{\text{f}}\;\left[ {{\text{r}}\;\left( {\text{i}} \right)} \right]$$ under the constraint $$\sum\limits_{i = 1}^N {{b^{ - r\left( i \right)}}\; \leqslant \;{b^t}}$$ where t is a non-negative integer and b is an integer \$#x2265; 2, is shown to have a very simple solution when the function f [r] is an exponential function of r and A (1), A (2), …., A (N) is any set of positive real numbers.

It is shown that when f [r] = ar where a is any real positivé number, a simple procedure based on the theory of uniquely decodable compact codes can be used to determine the integers r (1), r (2),… when t = 0. A minor modification on the already obtained values for the r’ s, will allow for other values of t.

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